Calculus Examples

$e^{- {(5 - 3 x)}^{2}}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - {(5 - 3 x)}^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $- {(5 - 3 x)}^{2}$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- {(5 - 3 x)}^{2}]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [- {(5 - 3 x)}^{2}]$

Step 1.3

Replace all occurrences of $u_{1}$ with $- {(5 - 3 x)}^{2}$ .

$e^{- {(5 - 3 x)}^{2}} \frac{d}{d x} [- {(5 - 3 x)}^{2}]$

Step 2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(5 - 3 x)}^{2}$ with respect to $x$ is $- \frac{d}{d x} [{(5 - 3 x)}^{2}]$ .

$e^{- {(5 - 3 x)}^{2}} (- \frac{d}{d x} [{(5 - 3 x)}^{2}])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 5 - 3 x$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $5 - 3 x$ .

$e^{- {(5 - 3 x)}^{2}} (- (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [5 - 3 x]))$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 2$ .

$e^{- {(5 - 3 x)}^{2}} (- (2 u_{2} \frac{d}{d x} [5 - 3 x]))$

Step 3.3

Replace all occurrences of $u_{2}$ with $5 - 3 x$ .

$e^{- {(5 - 3 x)}^{2}} (- (2 (5 - 3 x) \frac{d}{d x} [5 - 3 x]))$

Step 4

Differentiate.

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Step 4.1

Multiply $2$ by $- 1$ .

$e^{- {(5 - 3 x)}^{2}} (- 2 ((5 - 3 x) \frac{d}{d x} [5 - 3 x]))$

Step 4.2

By the Sum Rule, the derivative of $5 - 3 x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]$ .

$e^{- {(5 - 3 x)}^{2}} (- 2 (5 - 3 x) (\frac{d}{d x} [5] + \frac{d}{d x} [- 3 x]))$

Step 4.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$e^{- {(5 - 3 x)}^{2}} (- 2 (5 - 3 x) (0 + \frac{d}{d x} [- 3 x]))$

Step 4.4

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$e^{- {(5 - 3 x)}^{2}} (- 2 (5 - 3 x) \frac{d}{d x} [- 3 x])$

Step 4.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$e^{- {(5 - 3 x)}^{2}} (- 2 (5 - 3 x) (- 3 \frac{d}{d x} [x]))$

Step 4.6

Multiply $- 3$ by $- 2$ .

$e^{- {(5 - 3 x)}^{2}} (6 (5 - 3 x) \frac{d}{d x} [x])$

Step 4.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{- {(5 - 3 x)}^{2}} (6 (5 - 3 x) \cdot 1)$

Step 4.8

Multiply $6$ by $1$ .

$e^{- {(5 - 3 x)}^{2}} (6 (5 - 3 x))$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$e^{- {(5 - 3 x)}^{2}} (6 \cdot 5 + 6 (- 3 x))$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $6$ by $5$ .

$e^{- {(5 - 3 x)}^{2}} (30 + 6 (- 3 x))$

Step 5.2.2

Multiply $- 3$ by $6$ .

$e^{- {(5 - 3 x)}^{2}} (30 - 18 x)$